l -gain analysis for a class of hybrid systems with

2766 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER 2016

L2-Gain Analysis for a Class of Hybrid Systems WithApplications to Reset and Event-Triggered

Control: A Lifting ApproachW. P. M. H. Heemels, Fellow, IEEE, G. E. Dullerud, Fellow, IEEE, and A. R. Teel, Fellow, IEEE

Abstract—In this paper we study the stability and L2-gainproperties of a class of hybrid systems that exhibit linear flowdynamics, periodic time-triggered jumps and arbitrary nonlinearjump maps. This class of hybrid systems is relevant for a broadrange of applications including periodic event-triggered control,sampled-data reset control, sampled-data saturated control, andcertain networked control systems with scheduling protocols. Forthis class of continuous-time hybrid systems we provide new sta-bility and L2-gain analysis methods. Inspired by ideas from liftingwe show that the stability and the contractivity in L2-sense (mean-ing that the L2-gain is smaller than 1) of the continuous-timehybrid system is equivalent to the stability and the contractivityin �2-sense (meaning that the �2-gain is smaller than 1) of anappropriate discrete-time nonlinear system. These new character-izations generalize earlier (more conservative) conditions providedin the literature. We show via a reset control example and an even-t-triggered control application, for which stability and contractiv-ity in L2-sense is the same as stability and contractivity in �2-senseof a discrete-time piecewise linear system, that the new conditionsare significantly less conservative than the existing ones in theliterature. Moreover, we show that the existing conditions canbe reinterpreted as a conservative �2-gain analysis of a discrete-time piecewise linear system based on common quadratic stor-age/Lyapunov functions. These new insights are obtained by theadopted lifting-based perspective on this problem, which leads tocomputable �2-gain (and thusL2-gain) conditions, despite the factthat the linearity assumption, which is usually needed in the liftingliterature, is not satisfied.

Index Terms—Periodic event-triggered control (PETC), piece-wise affine (PWA), piecewise linear (PWL).

Manuscript received December 9, 2014; revised July 18, 2015; acceptedOctober 26, 2015. Date of publication November 20, 2015; date of currentversion September 23, 2016. This work was supported by the InnovationalResearch Incentives Scheme under the VICI grant “Wireless control systems:A new frontier in automation” 11382 awarded by the NWO (The NetherlandsOrganisation for Scientific Research) and STW (Dutch Technology Founda-tion), by grants NSA SoS W911NSF-13-0086 and AFOSR MURI FA9550-10-1-0573, by AFOSR under grants FA9550-12-1-0127 and FA9550-15-1-0155,and the National Science Foundation (NSF) under grants ECCS-1232035 andECCS-1508757. Recommended by Associate Editor C. Belta.

W. P. M. H. Heemels is with the Control System Technology Group,Department of Mechanical Engineering, Eindhoven University of Technology,Eindhoven, 5600 MB, The Netherlands (e-mail: [email protected]).

G. E. Dullerud is with the Mechanical Science and Engineering Department,and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801USA (e-mail: [email protected]).

A. R. Teel is with the Department of Electrical and Computer Engineering,University of California, Santa Barbara, CA 93106 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2015.2502422

I. INTRODUCTION

IN this paper we are interested in a class of hybrid systemsthat can be written in the framework of [1] as

d

dt

[ξτ

]=

[Aξ +Bw

1

], when τ ∈ [0, h] (1a)

[ξ+

τ+

]=

[φ(ξ)0

], when τ = h (1b)

z =Cξ +Dw. (1c)

The states of this hybrid system consist of ξ ∈ Rnξ and a

timer variable τ ∈ R≥0. The variable w ∈ Rnw denotes the

disturbance input and z the performance output. Moreover, A,B, C, D are constant real matrices of appropriate dimensions,h ∈ R>0 is a positive timer threshold, and φ : Rnξ → R

nξ

denotes an arbitrary nonlinear (possibly discontinuous) mapwith φ(0) = 0. Note that φ(0) = 0 guarantees that the set{[ξτ

]| ξ = 0 and τ ∈ [0, h]} is an equilibrium set of (1) in

absence of disturbances (w = 0).Interpreting the dynamics of (1) reveals that (1) has periodic

time-triggered jumps, i.e., jumps take place at times kh, k ∈ N

(when τ(0) = 0), according to a nonlinear jump map as givenby (1b). In between the jumps the system flows according tothe differential equations in (1a). This class of systems includesthe closed-loop systems arising from periodic event-triggeredcontrol (PETC) for linear systems [2], networked control withconstant transmission intervals and a shared networked requir-ing network protocols [3], [4], reset control systems [5]–[9]with periodically verified reset conditions, and sampled-datasaturated controls [10], as we will show in this paper. In allthe mentioned applications the function φ is a piecewise affine(PWA) map [11]. For other functions φ other application do-mains could be envisioned. Moreover, the results in this paperalso apply to set-valued mappings φ : Rnξ ⇒ R

nξ with φ(0) ={0} mutatis mutandis, see also [12]. However, for ease ofexposition, we restricted ourselves to single-valued functions.In any case, the modelling setup in (1) unifies several importantapplications in one framework, see also Section II below,which indicates the relevance of the class of systems understudy.

In this paper we are, besides showing the unifying modelingcharacter of the studied class of hybrid systems, interested inthe stability and L2-gain analysis from disturbance w to outputz for systems in the form (1). The L2-gain is an important

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HEEMELS et al.: L2-GAIN ANALYSIS FOR A CLASS OF HYBRID SYSTEMS WITH APPLICATIONS TO RESET AND EVENT-TRIGGERED CONTROL 2767

performance measure for many situations and the existingworks [1], [2], [10], [13] already focussed on obtaining upperbounds on this performance measure for the class of systems(1) with φ a piecewise linear (PWL) or piecewise affine (PWA)map. These works exploited timer-dependent Lyapunov/storagefunctions [14], [15] based on solutions to Riccati differentialequations. This resulted in LMI-based conditions leading toupper bounds on the L2-gain. In [13] improved conditions thatlead to better estimates of the L2-gain were derived using moreflexible Lyapunov functions. In the present paper, we provethat the LMI-based conditions obtained in [1], [2] and [10] canbe interpreted as �2-gain conditions using a common quadraticLyapunov/storage function [14] for discrete-time PWL or PWAsystems. Moreover, we will reveal that the LMI-based condi-tions obtained in [13] can be seen as an �2-gain analysis ofthe same PWL systems based on a special piecewise quadraticLyapunov/storage function. Interestingly, if these observationswould be particularized to linear sampled-data systems (i.e.,the case of (1) with φ a linear map), we would recover thewell-known lifted system approach from sampled-data controltheory, see, e.g., [16]–[21]. However, the classical lifting-basedapproach for sampled-data systems as in [16]–[21] focused onthe case of linear systems and controllers only and, in fact, thelinearity property was instrumental in the main developments.Clearly, linearity is a property not being satisfied for (1) whenφ is nonlinear, which is the case of interest in the current paper.Therefore, a new perspective is required on the problem at handif lifting-based techniques are to be exploited in a way leadingto verifiable conditions to determine the stability and L2-gainof systems of the form (1).

Despite the fact that the dynamics are nonlinear in (1), in thispaper we will establish that the stability and the γ-contractivity(in the sense that the L2-gain is smaller than γ) of the hybridsystem (1) is equivalent to the stability and the γ-contractivity(in the sense that the �2-gain is smaller than γ) of a specificdiscrete-time nonlinear system. As such, the L2-gain of thehybrid system (1) can be determined by studying the �2-gainfor discrete-time nonlinear systems. In the context of the PETC,networked control, saturated control and reset control applica-tions mentioned earlier and in which φ is a piecewise affine(PWA) mapping, this �2-gain can be closely upper boundedby employing piecewise quadratic Lyapunov/storage function[22], [23] for discrete-time PWA systems. As we will see, ournew method provides much better bounds on the L2-gain of(1) than the earlier results in [1], [2], [10], and [13] due tothe full equivalence between the stability and γ-contractivity(in L2-sense) of the hybrid system (1) and the stability andγ-contractivity (in �2-sense) of a specific discrete-time PWAsystem. Given the broad applicability of the hybrid model (1),these improved bounds might prove to be very valuable. This willbe illustrated in Section VII for two numerical examples in thecontext of reset and event-triggered control. Note that this papersignificantly extends our work reported in [12] as it providesfull proofs, a complete computational procedure (Section V),establishes the connections to existing techniques for stabilityand L2-gain analysis of (1) (Section VI) and presents a periodicreset control example that shows considerable improvementscompared to these existing techniques (all not in [12]). Finally,

an interesting observation is that, to the best of the authors’knowledge, the current paper is the first to employ lifting-liketechniques outside the linear domain in a manner that leadsto computable, easily verifiable conditions. These results canbe obtained by exploiting the structure in (1) having fixed(periodic) jump times and having a flow map that is linear forthe non-timer states ξ.

The remainder of this paper is organized as follows. InSection II we show how classes of networked control systems,reset control systems, periodic event-triggered controllers andsampled-data saturated control strategies can be captured inthe modelling framework based on (1). In Section III weintroduce the preliminaries and several definitions necessary toestablish the main results, which can be found in Section IV.The main result (Theorem IV.4) connects the internal stabilityand the contractivity in L2-sense of the hybrid system (1)to the internal stability and the contractivity in �2-sense ofa particular discrete-time nonlinear system. In Section V weindicate how particular matrices in the obtained discrete-timenonlinear system can be computed and how the internal stabilityand the �2-gain can be analyzed when φ in (1) is a PWL map.In Section VI we show how our lifting-based results connectto earlier results for the L2-gain analysis of the hybrid system(1) in [1], [2], [10], and [13]. In Section VII we show throughtwo numerical examples that the new conditions provided in thepresent paper lead to significantly less conservative conditionsthan the existing conditions in [1], [2], [10], and [13]. Finally,conclusions are stated in Section VIII. All technical proofs canbe found in the Appendix.

II. UNIFIED MODELLING FRAMEWORK

In this section, we will consider four different control ap-plications that can be cast in the hybrid system frameworkbased on (1).

A. Reset Control Systems

Reset control is a discontinuous control strategy proposedas a means to overcome the fundamental limitations of linearfeedback by allowing to reset the controller state, or subset ofstates, whenever certain conditions on its input and output aresatisfied, see e.g., [5]–[8]. In all afore-cited papers the resetcondition is monitored continuously, while recently in [9] it wasproposed to verify the reset condition at discrete-time instancesonly. In particular, at every sampling time tk = kh, k ∈ N, withsampling interval h > 0, it is decided whether or not a resettakes place. This type of reset controllers can be modelled as ahybrid system (1).

In order to show this, we study the control of a plant

{ddtxp = Apxp +Bpuu+Bpww

y = Cpxp

(2)

where xp ∈ Rnp denotes the state of the plant, u ∈ R

nu thecontrol input and y ∈ R

ny the plant output. The control system


is in the form of a reset controller of the type

d

dt

[xc

τ

]=

[Acxc +Bce

1

], when τ ∈ [0, h] (3a)

[x+c

τ+

]=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

[xc

0

], when τ = h and ξ�Qξ > 0[

Rcxc

0

], when τ = h and ξ�Qξ ≤ 0

(3b)

u =Ccxc +Dce (3c)

where xc ∈ Rnc denotes the continuous state of the controller

and x+c its value after a reset, Rc ∈ R

nc×nc is the reset matrixand e := r − y ∈ R

ny is the error between the reference signalr and the output y of the plant. Moreover, ξ := [x�

p x�c ]

� is anaugmented state vector containing plant and controller states.An example of a reset condition, originally proposed in [24]1

for the case nu = ny = 1, is based on the sign of the productbetween the error e and the controller input u. In particular, thereset controller (3) acts as a linear controller whenever its inpute and output u have the same sign, i.e., eu > 0, and it resets itsoutput otherwise. This reset condition can be represented, forthe case r = 0, in a general quadratic relation as in (3b), with

Q =

[Cp 0

−DcCp Cc

]� [0 −1−1 0

] [Cp 0

−DcCp Cc

]. (4)

The interconnection of the reset control system (3) and plant (2)can be written in the hybrid system format of (1) in which

A =

[Ap −BpuDcCp BpuCc

−BcCp Ac

], B =

[Bpw

0

]

and φ is a piecewise linear (PWL) map given for ξ ∈ Rnξ by

φ(ξ) =

{J1ξ, when ξ�Qξ > 0

J2ξ, when ξ�Qξ ≤ 0(5)

with J1 = Inξand J2 =

[Inp

00 Rc

].

B. Periodic Event-Triggered Control Systems

The second domain of application is event-triggered control(ETC), see e.g., [25] for a recent overview. ETC is a controlstrategy that is designed to reduce the usage of computation,communication and/or energy resources for the implementationof the control system by updating and communicating sensorand actuator data only when needed to guarantee specific sta-bility or performance properties. The ETC strategy that we con-sider in this paper combines ideas from periodic sampled-datacontrol and ETC, leading to so-called periodic event-triggeredcontrol (PETC) systems [2]. In PETC, the event-triggering con-dition is verified periodically in time instead of continuously asin standard ETC, see, e.g., [26], [27] and the references therein.

1Note that in [24], the reset condition is verified continuously instead of attimes tk = kh, k ∈ N only as in [9] and considered here.

Fig. 1. Schematic representation of an event-triggered control system.

Hence, at every sampling interval it is decided whether or notnew measurements and control signals need to be determinedand transmitted.

We consider again the plant (2), but now being controlled inan event-triggered feedback fashion using

u(t) = Kxp(t), for t ∈ R≥0 (6)

where xp ∈ Rnp is a left-continuous signal,2 given for t ∈

(tk, tk+1], k ∈ N, by

xp(t) =

{xp(tk), when ξ(tk)

�Qξ(tk) > 0

xp(tk), when ξ(tk)�Qξ(tk) ≤ 0

(7)

where ξ := [x�p x�

p ]� and tk, k ∈ N, are the sampling times,

which are periodic in the sense that tk = kh, k ∈ N with h > 0the sampling period. Fig. 1 shows a schematic representationof this PETC configuration. In this figure, xp denotes the mostrecently received measurement of the state xp available at thecontroller. Whether or not xp(tk) is transmitted is based on anevent generator (see (7)). In particular, if at time tk it holdsthat ξ�(tk)Qξ(tk) > 0, the current state xp(tk) is transmittedto the controller and xp and u are updated accordingly. If,however, ξ�(tk)Qξ(tk) ≤ 0, the current state information isnot sent to the controller and xp and u are kept the same for(at least) another sampling interval. In [2] it was shown thatsuch quadratic event-triggering conditions form a relevant classof event generators, as many popular event generators can bewritten in this form. For instance, in [26] events are generatedwhen ‖xp(tk)− xp(tk)‖ > ρ‖xp(tk)‖ (although verified con-tinuously instead of periodically), where ρ > 0. Clearly, thistriggering condition can be written in the quadratic form in (7)

by taking Q =

[(1− ρ2)I −I

−I I

]. In the numerical example

of Section VII-B another triggering condition will be shown.The complete model of the PETC system can be captured inthe hybrid system format of (1), by combining (2), (6) and

(7), where we obtain A =

[Ap BpuK0 0

], B =

[Bpw

0

], and

φ a PWL map as in (5) with J1 =

[Inp

0Inp

0

]and J2 = Inξ

.

Clearly, next to the case of static state-feedback controllersas in (6), one can also model dynamic output-feedback PETCcontrollers and output-based event-triggering conditions in theframework of (1), see [2] for more details.

2A signal x : R≥0 → Rn is called left-continuous, if for all t > 0,

lims↑t x(s) = x(t).


Fig. 2. Schematic representation of a networked control system.

C. Networked Control Systems

Networked control systems (NCSs) are control systems inwhich the control loops are closed over a real-time communi-cation network, see e.g., [28] for an overview and see Fig. 2 fora schematic. In this figure, y ∈ R

ny denotes the plant outputand y ∈ R

ny its so-called ‘networked’ version, i.e., the mostrecent output measurements of the plant that are available atthe controller. The control input is denoted by u ∈ R

nu andthe most recent control input available at the actuators is givenby u ∈ R

nu .If the transmission intervals are assumed to be constant

(equal to h) and the protocols, which determine the access tothe network, are assumed to be of a special type as studied, forinstance, in [3], [4], [29], and [30], these NCSs can be modeledin the framework (1). In order to show this, we consider plantsof the form (2) in which the control input u is replaced by itsnetworked version u. The output-feedback controller with statexc ∈ R

nc is assumed to be given by

d

dtxc = Acxc +Bcy u = Ccxc +Dcy (8)

although also a discrete-time controller can be used. Thenetwork-induced errors are defined as e = [e�y e�u ]

� with ey =y − y and eu = u− u, which describe the difference be-tween the most recently received information at the controller/actuators and the current value of the plant/controller output,respectively. The network is assumed to operate in a zero-orderhold (ZOH) fashion in between the updates of the values y andu, i.e., ˙y = 0 and ˙u = 0 between update times. We considerthe case where the plant is equipped with ny sensors and nu

actuators that are grouped into N communication nodes. Ateach transmission time tk = kh, k ∈ N, only one node σk ∈{1, 2, . . . , N} is allowed to communicate. Therefore, we obtainthe updates{

y(t+k)= Γy

σky(tk) +

(I − Γy

σk

)y(tk)

u(t+k)= Γu

σku(tk) +

(I − Γu

σk

)u(tk).

(9)

In (9), Γi := diag(Γyi ,Γ

ui ), i ∈ {1, . . . , N}, are diagonal ma-

trices given by Γi = diag(γi,1, . . . , γi,ny+nu), in which the

elements γi,j , with i ∈ {1, . . . , N} and j ∈ {1, . . . , ny}, areequal to one, if plant output yj is in node i and are zeroelsewhere, and elements γi,j+ny

, with i ∈ {1, . . . , N} and j ∈{1, . . . , nu}, are equal to one, if controller output uj is innode i and are zero elsewhere. Network protocols determinewhich node is allowed to access the network. The hybridframework (1) especially allows to study quadratic networkprotocols, see e.g., [3], [4], [29], [30], of the form

σk = arg mini∈{1,2,...,N}

ξ�(tk)Riξ(tk) (10)

in which Ri, i ∈ {1, . . . , N}, are certain given matrices andξ = [x�

p x�c e�y e�u ]

�. In fact, the well-known try-once-discard(TOD) protocol [29] belongs to this particular class of proto-cols. In this protocol, the node with the largest network-inducederror is granted access to the network in order to update its val-ues, which is defined by σk = argmaxi∈{1,...,N} ‖Γie(tk)‖2.

For simplicity, let us only consider two nodes (although theextension to N > 2 nodes can be done in a straightforwardfashion). The complete model of the NCS can be written in thehybrid system format of (1), by combining (2), (8), and (10) andtaking

A =

⎡⎢⎢⎣

Ap+BpuDcCp BpuCc BpuDc Bpu

BcCp Ac Bc 0−Cp(Ap+BpuDcCp) −CpBpuCc −CpBpuDc −CpBpu

−CcBcCp −CcAc −CcBc 0

⎤⎥⎥⎦

B =

⎡⎢⎢⎣

Bpw

0−CpBpw

0

⎤⎥⎥⎦

and φ is given as in (5) with3 Q = R2 −R1 and Ji =[I 00 I − Γi

]for i ∈ {1, 2}.

D. Sampled-Data Saturated Control Systems

In the setting of periodic sampled-data saturated control, seee.g., [10], the plant (2) is controlled by a sampled-data controllaw, such that

d

dtxp(t) = Apxp(t) +Bpuu(tk) +Bpww(t) (11)

for all t ∈ (tk, tk+1], k ∈ N. The control input u ∈ Rnu is sub-

ject to actuator saturation, having saturation levels u1, u2, . . . ,unu

> 0 on the respective input entries. Hence, the effectivecontrol signal for a state-feedback gain K = [K�

1 K�2 , . . . ,

K�nu

]� with Ki∈R1×np , i=1, 2, . . . , nu, is given for k ∈ N by

u(tk) = sat (Kxp(tk))

= [sat1 (K1xp(tk)) , sat2 (K2xp(tk)) , . . . ,

satnu(Knu

xp(tk))]�

with its i-th component given by ui(tk) = sati(Kixp(tk)) :=sign(Kixp(tk))min{ui, |Kixp(tk)|}, i = 1, 2, . . . , nu. Here,sign(a) denotes the sign of a scalar a.

The complete closed-loop model of the sampled-data satu-rated control system can now be written as a hybrid system(1) by taking the augmented state vector as ξ = [x�

p u�]�, and

using the matrices A =

[Ap Bpu

0 0

], B =

[Bbw

0

], and the

piecewise affine (PWA) map φ given by φ(ξ) =

[xp

sat(Kxp)

]for ξ ∈ R

nξ . In a similar manner also dynamic output-basedfeedback controllers with saturation and possibly PWA controlscan be captured in the hybrid model (1), see [10] for more details.

3In case ξ�R1ξ = ξ�R2ξ we assume for simplicity that node 2 gets accessto the network.


III. PRELIMINARIES

In this section we introduce preliminary definitions andnotational conventions.

For X,Y Hilbert spaces with inner products 〈·, ·〉X and〈·, ·〉Y , respectively, a linear operator U : X → Y is calledisometric if 〈Ux1, Ux2〉Y = 〈x1, x2〉X for all x1, x2 ∈ X . Wedenote by U ∗ : Y → X the (Hilbert) adjoint operator thatsatisfies 〈Ux, y〉Y = 〈x, U ∗y〉X for all x ∈ X and all y ∈ Y .Note that U being isometric is equivalent to U ∗U = I (orUU ∗ = I). The operator U is called an isomorphism if it isan invertible mapping. The induced norm of U (provided it isfinite) is denoted by ‖U‖X,Y = supx∈X\{0}(‖Ux‖Y /‖x‖X). Ifthe induced norm is finite we say that U is a bounded linearoperator. If X = Y we write ‖U‖X and if X,Y are clear fromthe context we use the notation ‖U‖. The image of U is writtenas im U and its kernel by kerU . An operator U : X → X withX a Hilbert space is called self-adjoint if U ∗ = U . A self-adjoint operator U : X → X is called positive semi-definite if〈Ux, x〉 ≥ 0 for all x ∈ X . Given a positive semi-definite U wesay that the bounded linear operator A : X → X is the squareroot of U if A is positive semi-definite andA2 = U . This squareroot exists and is unique, see Theorem 9.4-1 in [31]. We denoteit by U

12 .

To a Hilbert space X with inner product 〈·, ·〉X , we canassociate the Hilbert space �2(X) consisting of infinite se-quences x = (x0, x1, x2, . . .) with xi ∈ X , i ∈ N, satisfy-ing

∑∞i=0 ‖xi‖2X < ∞, and the inner product 〈x, y〉�2(X) =∑∞

i=0〈xi, yi〉X . We denote �2(Rn) by �2 when n ∈ N≥1 is clearfrom the context. We also use the notation �(X) to denote theset of all infinite sequences x = (x0, x1, x2, . . .) with xi ∈ X ,i ∈ N. Note that �2(X) can be considered a subspace of �(X).As usual, we denote by R

n the standard n-dimensionalEuclidean space with inner product 〈x, y〉 = x�y and norm|x| =

√x�x for x, y ∈ R

n. Ln2 ([0,∞)) denotes the set of

square-integrable functions defined on R≥0 := [0,∞) and

taking values in Rn with L2-norm ‖x‖L2

=√∫∞

0 |x(t)|2dtand inner product 〈x, y〉L2

=∫∞0 x�(t)y(t)dt for x, y ∈

Ln2 ([0,∞)). If n is clear from the context we also write L2.

We also use square-integrable functions on subsets [a, b] of R≥0

and then we write Ln2 ([a, b]) (or L2([a, b]) if n is clear from

context) with the inner product and norm defined analogously.The set Ln

2,e([0,∞)) consists of all locally square-integrablefunctions, i.e., all functions x defined on R≥0, such that foreach bounded domain [a, b] ⊂ R≥0 the restriction x|[a,b] iscontained in Ln

2 ([a, b]). We also will use the set of essentiallybounded functions defined on R≥0 or [a, b] ⊂ R≥0, which aredenoted by Ln

∞([0,∞)) or Ln∞([a, b]) with the norm given by

the essential supremum denoted by ‖x‖L∞ for an essentiallybounded function x. A function β : R≥0 → R≥0 is called a K-function if it is continuous, strictly increasing and β(0) = 0.

As the objective of the paper is to study the L2-gain andinternal stability of the system (1), let us first provide rigorousdefinitions of these important concepts.

Definition III.1: The hybrid system (1) is said to have anL2-gain from w to z smaller than γ if there exist a γ0 ∈ [0, γ)and a K-function β such that, for any w ∈ L2 and any initialconditions ξ(0) = ξ0 and τ(0) = h, the corresponding solution

to (1) satisfies ‖z‖L2≤ β(|ξ0|) + γ0‖w‖L2

. Sometimes we alsouse the terminology γ-contractivity (in L2-sense) if this prop-erty holds. Moreover, 1-contractivity is also called contractivity(in L2-sense).

Definition III.2: The hybrid system (1) is said to beinternally stable if there exists a K-function β such that,for any w ∈ L2 and any initial conditions ξ(0) = ξ0 andτ(0) = h, the corresponding solution to (1) satisfies ‖ξ‖L2

≤β(max(|ξ0|, ‖w‖L2

)).A few remarks are in order regarding this definition of in-

ternal stability. The requirement ‖ξ‖L2≤ β(max(|ξ0|, ‖w‖L2

))is rather natural in this context as we are working withL2-disturbances and investigate L2-gains. Indeed, just as inDefinition III.1, where a bound is required on the L2-norm ofthe output z (expressed in terms of a bound on |ξ0| and ‖w‖L2

),we require in Definition III.2 that a similar (though less strict)bound holds on the state trajectory ξ. Below we will showthat this property implies also global attractivity of the origin(i.e., limt→∞ ξ(t) = 0 for all w ∈ L2, ξ(0) = ξ0 and τ(0) = h)and also Lyapunov stability of the origin as we will also have‖ξ‖L∞ ≤ β(max(|ξ0|, ‖w‖L2

)), see Proposition IV.1 below. Inthe case where φ is positively homogeneous, i.e., φ satisfiesφ(λx) = λφ(x) for all x and all λ ≥ 0, and is continuous (orouter semicontinuous and locally bounded in the case φ is aset-valued mapping), it can be shown that the internal stabilityproperty is equivalent to the property that limt→∞ ξ(t) = 0 forany solution ξ corresponding to some initial conditions ξ(0) =ξ0 and τ(0) = h and zero disturbance w ≡ 0.

Consider the discrete-time system of the form

ξk+1 =χ(ξk, vk) (12a)

rk =ψ(ξk, vk) (12b)

with vk ∈ V , rk ∈ R, ξk ∈ Rnξ , k ∈ N, with V and R Hilbert

spaces, and χ : Rnξ × V → Rnξ and ψ : Rnξ × V → R.

Also for this general discrete-time system we formally intro-duce �2-gain specifications and internal stability.

Definition III.3: The discrete-time system (12) is said tohave an �2-gain from v to r smaller than γ if there exist aγ0 ∈ [0, γ) and a K-function β such that, for any v ∈ �2(V )and any initial state ξ0 ∈ R

nξ , the corresponding solution to(12) satisfies

‖r‖�2(R) ≤ β (‖ξ0‖) + γ0‖v‖�2(V ). (13)

Sometimes we also use the terminology γ-contractivity (in �2-sense) if this property holds. Moreover, 1-contractivity is alsocalled contractivity (in �2-sense).

Definition III.4: The discrete-time system (12) is said to beinternally stable if there is a K-function β such that, for any v ∈�2(V ) and any initial state ξ0 ∈ R

nξ , the corresponding solutionξ to (12) satisfies

‖ξ‖�2 ≤ β(max

(|ξ0|, ‖v‖�2(V )

). (14)

Note that this internal stability definition for the discrete-time system (12) parallels the continuous-time version inDefinition III.2. Moreover, since ‖ξ‖�∞ ≤ ‖ξ‖�2 and ‖ξ‖�2 <∞ implies limk→∞ ξk = 0, we also have global attractivity and


Lyapunov stability properties of the origin when the discrete-time system is internally stable.

The lemma below will be useful for later purposes. The proofcan be obtained by standard arguments and is therefore omitted.

Lemma III.5: Let Ha, Hb, and Hd be Hilbert spaces.Consider sequences a = {ak}k∈N ∈ �2(Ha), b = {bk}k∈N ∈�2(Hb), and d = {dk}k∈N with dk ∈ Hd, k ∈ N. If for α ≥ 0and β ≥ 0 it holds that ‖dk‖Hd

≤ α‖ak‖Ha+ β‖bk‖Hb

forall k ∈ N, then d ∈ �2(Hd) and ‖d‖�2(Hd) ≤ δα‖a‖�2(Ha) +δβ‖b‖�2(Hb) for some δα, δβ ≥ 0, i = 1, 2. Moreover, if 0 ≤α < 1 one can take 0 ≤ δα < 1.

IV. INTERNAL STABILITY AND L2-GAIN ANALYSIS

In this section we will analyze the L2-gain and the in-ternal stability of (1) using ideas from lifting [16]–[21]. Inparticular, we focus on the contractivity of the system (1) asγ-contractivity can be studied by proper scaling of the matricesC and D in (1).

To obtain necessary and sufficient conditions for the internalstability and the contractivity of (1), we will use a procedureconsisting of three main steps:

• In Section IV-A we apply lifting-based techniques to (1)(having finite-dimensional input and output spaces) lead-ing to a discrete-time system with infinite-dimensionalinput and output spaces (see (16) below). The internalstability and contractivity of both systems are equivalent.

• In Section IV-B we apply a loop transformation to theinfinite-dimensional system (16) in order to remove thefeedthrough term, which is the only operator in the sys-tem description having both its domain and range beinginfinite dimensional. This transformation is constructed insuch a manner that the internal stability and contractivityproperties of the system are not changed. This step iscrucial for translating the infinite-dimensional system toa finite-dimensional system in the last step.

• In Section IV-C the loop-transformed infinite-dimensionalsystem is converted into a discrete-time finite-dimensionalnonlinear system (again without changing the stabilityand the contractivity properties of the system). Due tothe finite dimensionality of the latter system, stability andcontractivity in �2-sense can be analyzed, for instance,using well-known Lyapunov-based arguments. We willelaborate on these computational aspects in Section V.

These three steps lead to the main result as formulatedin Theorem IV.4, which states that the internal stability andcontractivity (in L2-sense) of (1) is equivalent to the inter-nal stability and contractivity (in �2-sense) of a discrete-timefinite-dimensional nonlinear system. All proofs of the technicalresults can be found in the Appendix.

A. Lifting

To study contractivity, we introduce the lifting operator W :L2,e[0,∞)→�(K) with K=L2[0, h] given for w∈L2,e[0,∞) by W (w) = w = (w0, w1, w2, . . .) with

wk(s) = w(kh + s) for s ∈ [0, h] (15)

for k ∈ N. Obviously, W is a linear isomorphism mappingL2,e[0,∞) into �(K) and, moreover, W is isometric as amapping from L2[0,∞) to �2(K). Using this lifting operator,we can rewrite the model in (1) as

ξk+1 = Aξ+k + Bwk (16a)

ξ+k =φ(ξk) (16b)

zk = Cξ+k + Dwk (16c)

in which ξ0 is given and ξk = ξ(kh−)= lims↑kh ξ(s), k ∈ N≥1,and ξ+k = ξ(kh+) = lims↓kh ξ(s) = ξ(kh) (assuming that ξ iscontinuous from the right) for k ∈ N, and w = (w0, w1, w2,. . .) = W (w)∈ �2(K) and z = (z0, z1, z2, . . .)=W (z)∈�(K).Here we assume in line with Definition III.1 that τ(0) = h in(1). Moreover

A : Rnξ → Rnξ B : K → R

nξ

C : Rnξ → K D : K → K

are given for x ∈ Rnξ and ω ∈ K by

Ax = eAhx (17a)

Bω =

h∫0

eA(h−s)Bω(s)ds (17b)

(Cx)(θ) =CeAθξ (17c)

(Dω)(θ) =

θ∫0

CeA(θ−s)Bω(s)ds+Dω(θ) (17d)

where θ ∈ [0, h].By writing the solution of (1) explicitly, comparing to the

formulas (17) and using that W is an isometric isomorphism, itfollows that (16) is contractive if and only if (1) is contractive.In fact, we have the following proposition.

Proposition IV.1: The following statements hold:

• The hybrid system (1) is internally stable if and only if thediscrete-time system (16) is internally stable.

• The hybrid system (1) is contractive if and only if thediscrete-time system (16) is contractive.

• Moreover, in case (1) is internally stable, it also holds thatlimt→∞ ξ(t) = 0 and ‖ξ‖L∞ ≤ β(max(|ξ0|, ‖w‖L2

)) forall w ∈ L2, ξ(0) = ξ0 and τ(0) = h.

B. Removing the Feedthrough Term

Following [17] we aim at removing the feedthrough operatorD as this is the only operator with both its domain and rangebeing infinite dimensional. Removal can be accomplished byusing an operator-valued version of Redheffer’s lemma, seeLemma 5 in [17]. The objective is to obtain a new system(without feedthrough term) and new disturbance inputs vk ∈ K,new state ξk ∈ R

nξ and new performance output rk ∈ K, k ∈N, given by

ξk+1 = Aξ+k + Bvk (18a)

ξ+k =φ(ξk) (18b)

rk = Cξ+k (18c)


such that (16) is internally stable and contractive if and onlyif (18) is internally stable and contractive. To do so, we firstobserve that a necessary condition for the contractivity (16) isthat ‖D‖K < 1. Indeed, ‖D‖K ≥ 1 would imply that for any0 ≤ γ0 < 1 there is a w0 ∈ K \ {0}with ‖Dw0‖K ≥ γ0‖w0‖K,which, in turn, would lead for the system (16) with ξ0 = 0and thus ξ+0 = 0 and disturbance sequence (w0, 0, 0, . . .) toa contradiction with the contractivity of (16). The followingproposition will be useful in the sequel. It can be establishedbased on [31].

Proposition IV.2: Consider a linear bounded operator D :H → H with H a real Hilbert space and let ‖D‖H < 1. Thenwe have the following results.

1) ‖D∗D‖H = ‖D‖2H = ‖DD∗‖H = ‖D∗‖2H .2) The operators I − D∗D and I − DD∗ are invertible,

bounded, and positive semi-definite.

3) The operators (I−D∗D)−1

, (I−DD∗)−1

, (I−D∗D)1/2

,

(I − DD∗)1/2

, (I − D∗D)−(1/2)

, and (I − DD∗)−(1/2)

are invertible, bounded, and positive semi-definite.4) For l ∈ Z it holds that

(I − D∗D)l 12 D∗ = D∗(I − DD∗)

l 12 and

(I − DD∗)l 12 D = D(I − D∗D)

l 12 .

Consider now the linear bounded operator Θ : K ×K →K×K given by

Θ =

(−D (I − DD∗)

12

(I − D∗D)12 D∗

). (19)

The operatorΘ is unitary in the sense that Θ∗Θ = I andΘΘ∗ =I , see Theorem 6 in [17], and thus for all u, v ∈ K ×K we have

〈Θu,Θv〉K×K = 〈u, v〉K×K and ‖Θu‖K×K = ‖u‖K×K (20)

implying that Θ is an isometric isomorphism. This operator will

be used to transform, for each k ∈ N,

(wk

zk

)of (16) into

(vkrk

)according to the following equality:(

rkwk

)= Θ

(vkzk

). (21)

In fact, given

(wk

zk

)we can uniquely solve (21) leading to

(vkrk

)=

((I − D∗D)

− 12 −(I − D∗D)

− 12 D∗

−D(I − D∗D)− 1

2 (I − DD∗)− 1

2

)(wk

zk

).

(22)

Conversely, when

(vkrk

)is given, we can uniquely solve (21) to

obtain(wk

zk

)=

((I − D∗D)

− 12 (I − D∗D)

− 12 D∗

(I − DD∗)− 1

2 D (I − DD∗)− 1

2

)(vkrk

).

(23)

Hence, the mappings in (23) and (22) are both isomorphismsand (22) is the inverse of (23) and vice versa. Combining (21)with (16c) gives

wk =(I − D∗D)12 vk + D∗zk

=(I − D∗D)12 vk + D∗

[Cξ+k + Dwk

]. (24)

Solving for wk gives

wk =(I − D∗D)− 1

2 vk + (I − D∗D)−1D∗Cξ+k

=(I − D∗D)− 1

2 vk + D∗(I − DD∗)−1Cξ+k (25)

and leads to the system (18) with bounded linear operators

A : Rnξ → Rnξ B : K → R

nξ C : Rnξ → K

given by

A = A+ BD∗(I − DD∗)−1C (26a)

B = B(I − D∗D)− 1

2 (26b)

C =(I − DD∗)− 1

2 C. (26c)

Note that C follows from the calculations:

rk = −Dvk + (I − DD∗)12 zk

= −Dvk + (I − DD∗)12

[Cξ+k + D(I − D∗D)

− 12 vk

+ DD∗(I − DD∗)−1Cξ+k

]=(I − DD∗)

12

[I + DD∗(I − DD∗)

−1]Cξ+k

=(I − DD∗)− 1

2 Cξ+k (27)

where we used in the last equality that I+DD∗(I−DD∗)−1

=

(I − DD∗)−1

.Based on the above developments we can establish now the

following result.Theorem IV.3: Consider the system (16) with A, B, C, and

D as in (17) and assume ‖D‖K < 1. Consider also system (18)with A, B, and C as in (26). Then the following hold.

1) Let ξ0 and w = (w0, w1, . . .) ∈ �(K) be given and lead-ing to a state sequence {ξk}k∈N and output sequence z =(z0, z1, . . .) ∈ �(K) for system (16). Then there existsv = (v0, v1, . . .) ∈ �(K) such that (18) with initial stateξ0 = ξ0 leads to the state trajectory {ξk}k∈N and outputsequence r = (r0, r1, . . .) ∈ �(K) satisfying for k ∈ N

ξk = ξk and (28)

‖rk‖2K − ‖vk‖2K = ‖zk‖2K − ‖wk‖2K. (29)

2) Let ξ0 and v = (v0, v1, . . .) ∈ �(K) be given and leadingto a state sequence {ξk}k∈N and output sequence r =(r0, r1, . . .) ∈ �(K) for system (18). Then there existsw = (w0, w1, . . .) ∈ �(K) such that (16) with initial state


ξ0 = ξ0 leads to the state trajectory {ξk}k∈N and out-put sequence z = (z0, z1, . . .) ∈ �(K) satisfying (28) and(29) for k ∈ N.

3) Internal stability and contractivity of (16) are equivalentto internal stability and contractivity of (18).

C. From Infinite-Dimensional to Finite-Dimensional Systems

The system (18) is still an infinite-dimensional system, al-though the operators A, B, and C have finite rank and thereforehave finite-dimensional matrix representations. Following (andslightly extending) [17] we now obtain the following result.

Theorem IV.4: Consider system (1) and its lifted version(16) with ‖D‖K < 1. Define the discrete-time nonlinear system

ξk+1 =Adφ(ξk) +Bdvk (30a)

rk =Cdφ(ξk) (30b)

with

Ad = A+ BD∗(I − DD∗)−1C (31a)

and Bd ∈ Rnξ×nv and Cd ∈ R

nr×nξ are chosen such that

BdB�d = BB∗ = B(I − D∗D)

−1B∗ and

C�d Cd = C∗C = C∗(I − DD∗)

−1C. (31b)

The system (1) is internally stable and contractive if and only ifthe system (30) is internally stable and contractive.

Hence, this theorem states that under the assumption‖D‖K < 1 (which is a necessary condition for contractivity of(1)) the internal stability and contractivity (in L2-sense) of (1) isequivalent to the internal stability and contractivity (in �2-sense)of a discrete-time finite-dimensional nonlinear system given by(30). In the next section we will show how the matrices Ad, Bd,andCd in (30) can be constructed, how the condition ‖D‖K < 1can be tested, and how internal stability and contractivity canbe verified for the system (30) in case the nonlinear mappingφ is piecewise linear as in (5), which is relevant for severalapplications mentioned in Section II.

V. COMPUTATIONAL CONSIDERATIONS

In this section we demonstrate how the discrete-time system(30) provided in Theorem IV.4 can be computed and how theinternal stability and contractivity analysis can be carried outfor the discrete-time system (30) when φ is PWL as in (5).

A. Computing the Discrete-Time Nonlinear System

To explicitly compute the discrete-time system (30) providedin Theorem IV.4 we need to determine the operators BD∗(I −DD∗)−1C, B(I − D∗D)

−1B∗, and C∗(I − DD∗)

−1C to ob-

tain the triple (Ad, Bd, Cd) in (30). For self-containednesswe recall the procedure proposed in [32] to compute them,assuming throughout that ‖D‖K < 1.

First we recall the tests given in [16] and [32] to verify‖D‖K < 1, which is a necessary condition for the contractivity

of (1). The condition ‖D‖K < 1 is normally verified usingLemma 3.2 in [32] or Theorem 13.5.1 in [16]. In fact, inthese references ‖D‖K < 1 is shown to be equivalent to ‖D‖ =√λmax(D�D) < 1 and Qγ

11(h) being invertible for all γ ≥ 1

with Qγ(t) :=

(Qγ

11(t) Qγ12(t)

Qγ21(t) Qγ

22(t)

)= eE

γt for t ∈ R and

Eγ :=

[−A� − γ−2C�DMγB� −C�LγC

γ−2BMγB� A+ γ−2BMγD�C

](32)

where Mγ :=(I− γ−2D�D)−1

and Lγ := (I − γ−2DD�)−1

.Alternative tests can also be found in [33]. In Theorem VI.3below we will present another equivalent test (given inAssumption VI.1), which has some computational advantagesand is useful for constructing Lyapunov/storage functions for(16) proving the contractivity and internal stability in specificcases (see Section VI).

The procedure in [32] to find Ad, Bd, and Cd boils down tocomputing Q(h) := Q1(h), which then leads to

Ad = Q11(h)−� (33)

and Bd and Cd are matrices satisfying

BdB�d =Q21(h)Q11(h)

−1

C�d Cd = −Q11(h)

−1Q12(h) (34)

see [32] for the details. This provides the matrices needed forexplicitly determining the discrete-time nonlinear system (30)for which the internal stability and contractivity tests need to becarried out. In the next section we show which computationaltools can be used to carry out these tests for the situation whereφ is a PWL map as in (5).

B. Stability and Contractivity of Discrete-Time PWL Systems

For several important applications, including the reset andevent-triggered control systems mentioned in Section II-A andB, respectively, the nonlinear mapping φ in the hybrid system(1) is PWL as specified in (5). As a consequence, the system(30) in Theorem IV.4 particularizes in this case to the discrete-time system

ξk+1 =

{A1ξk +Bdvk, when ξ�k Qξk > 0

A2ξk +Bdvk, when ξ�k Qξk ≤ 0(35a)

rk =

{C1ξk, when ξ�k Qξk > 0

C2ξk, when ξ�k Qξk ≤ 0(35b)

k ∈ N, with Ai = AdJi, and Ci = CdJi, i = 1, 2.To guarantee the internal stability and contractivity of a

discrete-time PWL system as in (35) (in order to guaranteethese properties for the hybrid system (1) using Theorem IV.4),an effective approach is to use versatile piecewise quadraticLyapunov/storage functions [22], [23] of the form

V (ξ)=ξ�Piξ with i=min {j ∈ {1, . . . , N}|ξ∈Ωj} (36)


based on the regions

Ωi := {ξ ∈ Rnξ | ξ�Xiξ ≥ 0}, i ∈ {1, . . . , N} (37)

in which the symmetric matrices Xi, i ∈ {1, . . . , N}, are suchthat {Ω1,Ω2, . . . ,ΩN} forms a partition of Rnξ , i.e., ∪N

i=1Ωi =R

nξ and the intersection of Ωi ∩ Ωj is of zero measure forall i, j ∈ {1, . . . , N} with i �= j. Moreover, we assume that{ξ ∈ R

nξ | ξ�Qξ ≤ 0} =⋃N1

i=1 Ωi and {ξ ∈ Rnξ | ξ�Qξ ≥

0} =⋃N

i=N1+1 Ωi for some N1 < N .To establish contractivity of (35) we will use the dissipation

inequality [14], [15]

V (ξk+1)− V (ξk) ≤ −r�k rk + v�k vk, k ∈ N (38)

and require that it holds along the trajectories of the system(35). This translates into sufficient LMI-based conditions forstability and contractivity using three S-procedure relaxations[34], as formulated next.

Theorem V.1: Let N1 < N hold. Suppose that there ex-ist matrices Pi = P�

i , i ∈ {1, . . . , N}, and scalars μi,j ≥ 0,βi,j ≥ 0 and κi ≥ 0, i, j ∈ {1, . . . , N}, satisfying[Pi−μi,jXi−βi,jA

�1XjA1−C�

1C1−A�1PjA1 −A�

1PjBd

� I−B�dPjBd

]�0

(39a)

for all i ∈ {N1 + 1, . . . , N}, j ∈ {1, . . . , N}, and[Pi−μi,jXi−βi,jA

�2XjA2−C�

2 C2−A�2PjA2 −A�

2PjBd

� I−B�d PjBd

]�0

(39b)

for all i ∈ {1, . . . , N1}, j ∈ {1, . . . , N}, and

Pi − κiXi � 0, for all i ∈ {1, . . . , N}. (39c)

Then the discrete-time PWL system (35) is internally stable andcontractive.

Two comments are in order regarding this theorem. Firstly,note that due to the strictness of the LMIs (39) we guaranteethat the �2-gain is strictly smaller than 1, which can be seenfrom appropriately including the strictness into the dissipativityinequality (38). Moreover, due to the strictness of the LMIs wealso guarantee internal stability. Secondly, the LMI conditionsof Theorem V.1 are obtained by performing a contractivityanalysis on the discrete-time PWL system (35) using all pos-sible S-procedure relaxations, i.e.,

(1) require that ξ�Piξ is positive only when ξ ∈ Ωi \ {0},i.e., in (39c) we have Pi − κiXi � 0 for all i ∈ {1, . . . ,N} and κi ≥ 0;

(2) use a relaxation related to the current time instant, i.e.,if V (ξk) = ξ�k Piξk, then it holds that ξ�k Xiξk ≥ 0 (thiscorresponds to the terms μi,jXi in (39a) and (39b));

(3) use a relaxation related to the next time instant, i.e., ifV (ξk+1) = ξ�k+1Pjξk+1, then it holds that ξ�k+1Xj

ξk+1 ≥ 0 (this corresponds to the terms −βi,jA�1XjA1

and −βi,jA�2XjA2 in (39a) and (39b)).

Theorem V.1 can be used to guarantee the internal stabilityand contractivity of (35) and hence, the internal stability and

contractivity for the hybrid system (1) with φ given by (5). Inthe next section we will rigorously show that these results formsignificant improvements with respect to the earlier conditionsfor contractivity of (1) presented in [1], [2], [10], and [13].In Section VII we also illustrate this improvement using twonumerical examples.

Remark V.2: Similar techniques as above can also beapplied to the stability and (γ-)contractivity of sampled-datasaturated control systems described in Section II-D in whichthe mapping φ in the hybrid system (1) is PWA and the discrete-time nonlinear system (30) particularizes to a PWA system. Inthis case also piecewise quadratic Lyapunov/storage functionscan be used leading to LMI-based conditions, see [22], [23].

VI. CONNECTIONS TO AN EXISTING

LYAPUNOV-BASED APPROACH

In this section we will recall the LMI-based conditionsfor analyzing the stability and contractivity analysis for (1)provided in [2], [10], and [13], focussing on the case where φ isPWL as given in (5), and show the relationship to the conditionsobtained in the present paper. This will also reveal that theconditions in this paper are (significantly) less conservative.

We follow here the setup discussed in [2], which is basedon using a timer-dependent storage function V (ξ, τ), see [14],satisfying:

d

dtV ≤ −z�z + w�w (40)

during the flow (1a), and

V (J1ξ, 0) ≤V (ξ, h), for all ξ with ξ�Qξ > 0 (41a)

V (J2ξ, 0) ≤V (ξ, h), for all ξ with ξ�Qξ ≤ 0 (41b)

during the jumps (1b) with φ as in (5). From these conditions,we can guarantee that the L2-gain from w to z is smaller thanor equal to 1, see, e.g., [30]. In fact, in [2] V (ξ, τ) was chosenin the form

V (ξ, τ) = ξ�P (τ)ξ (42)

with P (·) the solution to the Riccati differential equation

d

dτP =−A�P−PA−C�C−(PB+C�D)M(B�P+D�C)

(43)provided the solution exists on [0, h], in which M :=

(I −D�D)−1

exists and is positive definite as we assume,as before, that 1 > λmax(D

�D). As shown in the proof of[2, Theorem III.2], this choice for the matrix function P impliesthe “flow condition” (40). The “jump condition” (41) is guaran-teed in [2] by LMI-based conditions that lead to a proper choiceof the boundary value Ph := P (h). To formulate the result of[2], we introduce the Hamiltonian matrix

H :=

[A+BMD�C BMB�

−C�LC −(A+BMD�C)�

](44)


with L := (I −DD�)−1

, which is positive definite, since 1 >λmax(D

�D) = λmax(DD�). In addition, we introduce thematrix exponential

F (τ) := e−Hτ =

[F11(τ) F12(τ)F21(τ) F22(τ)

](45)

allowing us to provide the explicit solution to the Riccatidifferential equation (43), yielding

P (0) = (F21(h) + F22(h)Ph) (F11(h) + F12(h)Ph)−1 (46)

provided that the solution (46) is well defined on [0, h], see,e.g., [35, Lem. 8.2]. To guarantee this, in [2] the followingassumption was used.

Assummption VI.1: λmax(D�D) < 1 and F11(τ) is invert-

ible for all τ ∈ [0, h].Let us also introduce the notation F11 := F11(h), F12 :=

F12(h), F21 := F21(h), and F22 := F22(h), and a matrix Sthat satisfies SS� = −F−1

11 F12. In [2] the following LMIswere derived guaranteeing (41) by using (46). Note that weapplied here an additional Schur complement compared to theequivalent LMIs formulated in [2].

Proposition VI.2: Consider the hybrid system (1) and letAssumption VI.1 hold. Suppose that there exist a matrix Ph �0, and scalars μi ≥ 0, i ∈ {1, 2}, such that for i ∈ {1, 2}[Ph+(−1)iμiQ−J�i F21F

−111 Ji−J�i F−�

11 PhF−111 Ji J�i F

−�11 PhS

� I− S�PhS

]�0.

(47)Then, the hybrid system (1) is internally stable and has anL2-gain from w to z smaller than or equal to 1.

In the spirit of Section V-B, it is not difficult to see that theLMI-based conditions in this proposition can be shown to beequivalent to a conservative check of the �2-gain being smallerthan or equal to 1 for the discrete-time piecewise linear (PWL)system

ξk+1 =

{F−111 J1ξk + Swk if ξ�k Qξk > 0

F−111 J2ξk + Swk if ξ�k Qξk ≤ 0

(48a)

zk =

{CJ1ξk if ξ�k Qξk > 0

CJ2ξk if ξ�k Qξk ≤ 0(48b)

k ∈ N, where S and C satisfy

SS� = −F−111 F12 and C�C := F21F

−111 . (49)

In particular, the stability and contractivity tests inProposition VI.2 use a common quadratic storage functionand only one of the S-procedure relaxations discussed inSection V-B (only (ii) is used). In addition to this new perspec-tive on the results in [1], [2] and [10], a strong link can be es-tablished between the existing LMI-based conditions describedin Proposition VI.2 and the lifting-based conditions obtained inthis paper, as formalized next.

Theorem VI.3: The following statements are true:

1) Assumption VI.1 is equivalent to ‖D‖K < 1.2) If ‖D‖K < 1, then the PWL system (35) and the PWL

system (35) are essentially the same in the sense thatF−111 = Ad, SS� = BdB

�d , and C�C = C�

d Cd.

Moreover, if the hypotheses of Proposition VI.2 hold and theregions4 in (37) are chosen such that for each i = 1, 2, . . . , Nthere is a ξi ∈ R

nξ such that ξ�i Xiξi > 0, then ‖D‖K < 1 andthe hypotheses of Theorem V.1 hold.

This theorem reveals an intimate connection between theresults obtained in [1], [2] and [10] and the new lifting-basedresults obtained in the present paper. Indeed, as already men-tioned, the LMI-based conditions in [1], [2] and [10] as formu-lated in Proposition VI.2 boil down to an �2-gain analysis ofa discrete-time PWL system (35) based on a quadratic storagefunction using only a part of the S-procedure relaxations pos-sible (only using (ii), while the S-procedure relaxations (i)and (iii) mentioned at the end of Section V-B are not used).Moreover, Theorem VI.3 shows that the new lifting-basedresults using Theorem V.1 and Theorem IV.4 never provideworse estimates of the L2-gain of (1) than the existing resultsas formulated in Proposition VI.2. In fact, since the stabilityand contractivity conditions based on (35) can be carried outbased on more versatile piecewise quadratic storage functionsand more (S-procedure) relaxations (see Theorem V.1), the newconditions are typically significantly less conservative than theones obtained in [1], [2], [10]. These benefits will be demon-strated quantitatively in the next section using two numericalexamples.

Remark VI.4: To emphasize, note that in [1], [2], [10]the connection to an �2-gain analysis of a PWL system wasnot uncovered. In the present paper, we do not only uncoverthis connection, but we show even the complete equivalencebetween the stability and contractivity of (1) and the stabilityand the stability and contractivity of (35), and not only forPWL maps φ, but for arbitrary nonlinear (even set-valued)maps φ.

Remark VI.5: The above uncovered connection also showsthat the approach in [1], [2] and [10], which does not resortto lifting-based arguments or infinite-dimensional systems, pro-vides an alternative route in the linear sampled-data context toobtain the equivalence of the stability and the contractivity of(1) with φ linear and the stability and the contractivity of a par-ticular discrete-time linear system. Moreover, the result in [1],[2] and [10] also provides as a byproduct a Lyapunov/storagefunction proving the internal stability and contractivity (in thesense of dissipativity with the supply rate −z�z + w�w bysatisfying (40) and (41)) for the system (1).

Remark VI.6: Note that we also improve with respect to ourrecent Lyapunov-based conditions in [13], which establishedonly upper bounds on the L2-gain of (1) by using piecewisequadratic Lyapunov/storage functions of the form (36) and onlyone relaxation as in (ii). The full equivalence as proven inTheorem IV.4 was not obtained in [13].

VII. NUMERICAL EXAMPLES

In this section we illustrate the improvement of the presentedtheory with respect to the existing literature using two numeri-cal examples.

4This condition implies that each region has a non-empty interior therebyavoiding redundant regions of zero measure.


Fig. 3. L2-gain as a function of the pole β of the FORE.

A. A Periodic Reset Control Application

The first example is inspired by [8] and studies a reset controlapplication. In this example, the plant consists of an integratorsystem of the form (2) with [Ap Bpu Bpw Cp] =[0 1 1 1], which is controlled by a periodic First-Order Reset Element (FORE) of the form (3) with[Ac Bc

Cc Dc

]=

[−β 11 0

], Rc = 0, and sampling interval h =

0.01. Q is given as in (4).To apply the lifting-based results in this paper to analyze

the stability and L2-gain (γ-contractivity) of the resultingclosed-loop reset control system in the form (1) we have todetermine the contractivity of the discrete-time PWL linearsystem (35) for various scaled values of C and D (next tochecking ‖D‖K < 1). We will perform such an analysis basedon the method discussed in Section V-B using the piecewisequadratic Lyapunov/storage function (36) To do so, we exploita partition of the state-space into N regions Ωi, i ∈ {1, . . . , N}inspired by [7], [36] and based on defining the vectors φi =[− sin(θi) cos(θi)]

� for θi = iπ/N , i ∈ {0, 1, . . . , N}, with Nan even number. We define the matrices Si = φi(−φi−1)

� +φi−1(−φi)

�, which lead to the symmetric matrices

Xi =

[Cp 0

−DcCp Cc

]�Si

[Cp 0

−DcCp Cc

], i∈{1, . . . , N}

(50)providing the state space partition as discussed in Section V-B,see also [7], [36] for more details. In the remainder of thisexample, we select N1 = 5, N = 10.

In Fig. 3, the upper bounds on the L2-gain of the closed-loopsystem are presented as a function of the pole β of the FOREfor both the existing and the new approaches. The thick solid(blue) line is included for comparison reasons and is obtainedby the LMI conditions of [36, Theorem 3] (in which essentiallyh = 0 and the reset conditions are checked continuously insteadof periodically). The dashed (magenta) curve is obtained bythe conditions of [2, Theorem III.2] (Proposition VI.2 of thispaper). The thin(ner) solid (green) line is obtained by the con-ditions of [13, Theorem IV.2], see also Remark VI.6. Finally,the dash-dotted (red) line is obtained by our new conditions inTheorem V.1. Due to Theorem VI.3 it is guaranteed that thenew conditions in Theorem V.1 would never be worse than

Fig. 4. Upper bound of the L2-gain as a function of the triggering parameter ρ.The solid (blue) line is based on [2], while the dashed (red) line uses the newresults presented in this paper.

the results of [2, Theorem III.2] (Proposition VI.2). However,due to several additional relaxations in Theorem V.1, we expectsignificant improvements. The displayed curves indeed confirmthis expectation and show that the new results in Theorem V.1provide a significant improvement compared to both the exist-ing approaches. To stress this further, observe that for β > 0 theapproach in [2, Theorem III.2] could not even establish a finiteL2-gain, while the new approach presented here does lead tosuch guarantees.

B. A Periodic Event-Triggered Control Application

In this example the plant

d

dtxp =

[1 2−2 1

]xp +

[01

]u+

[11

]w (51)

will be controlled using a PETC strategy specified by (6), (7),in which K = [−0.45 − 3.25]. At sampling times tk = kh,k ∈ N, with h = 0.19, we will transmit the state xp(tk) tothe controller and update the control action when ‖Kxp(tk)−Kxp(tk)‖ > ρ‖Kxp(tk)‖ with ρ ≥ 0. This PETC setup corre-sponds to

Q =

[(1 − ρ2)K�K −K�K

−K�K K�K

](52)

in the function φ given in (5) for (1).To study the stability and the L2-gain of the resulting closed-

loop system in the form (1) we follow the same procedure basedon the method discussed in Section V-B and the piecewisequadratic Lyapunov/storage function (36) as in the previousexample. The partition of the state-space into N regions Ωi,i ∈ {1, . . . , N} as in (37) is again based on the ideas in [7],[36], where we take N1 = 1 and N = 4 for various values ofρ ≥ 0. This results in Fig. 4. Also the upper bounds on theL2-gain of (1) corresponding to the sufficient conditions ob-tained in the earlier works [1], [2], [10] are provided. InFig. 4 we observe that also for this PETC application the newconditions lead to significantly better bounds than the existingones.

VIII. CONCLUSION

In this paper we studied internal stability and L2-gain prop-erties of a class of hybrid systems that exhibit linear flow


dynamics, periodic time-triggered jumps and arbitrary nonlin-ear jump maps. We showed the relevance of this class of hybridsystems by explaining how a broad range of applications inevent-triggered control, sampled-data reset control, sampled-data saturated control, and networked control can be capturedin this unifying modelling framework. In addition, we derivednovel conditions for both the internal stability and the contrac-tivity (in terms of L2-gains) for these dynamical systems. Inparticular, we provided a lifting-based approach that revealedthat the stability and the contractivity of the continuous-timehybrid system is equivalent to the stability and the contractivity(now in terms of �2-gains) of an appropriate discrete-timenonlinear system. These new lifting-based characterizationsgeneralize earlier (more conservative) conditions provided inthe literature and we showed via a reset control example anda periodic event-triggered control application, for which theL2-gain analysis reduces to an �2-gain analysis of discrete-time piecewise linear systems, that the new conditions aresignificantly less conservative than the existing ones. Moreover,we showed that the existing conditions can be reinterpretedas a conservative �2-gain analysis of a discrete-time piecewiselinear system based on common quadratic storage/Lyapunovfunctions. These new insights were obtained by the adoptedlifting-based perspective on this problem, which leads to com-putable �2-gain (and thus L2-gain) conditions, despite the factthat the linearity assumption, which is usually needed in thelifting literature, is not satisfied for this class of systems.

APPENDIX

Proof of Proposition IV.1: As already stated beforethe proposition, it is straightforward to see that contractivityis equivalent for both systems. To show that internal stabil-ity carries over, assume first that (1) is internally stable andconsider w ∈ �2(K) and initial state ξ0 for (16) leading tothe discrete-time state trajectory {ξk}k∈N (and correspondinglyalso to {ξ+k }k∈N). Also consider the corresponding “unlifted”disturbance version w = W−1(w) ∈ L2, and the trajectoryξ ∈ L2 of (1) corresponding to ξ(0) = ξ0 and τ(0) = h. Inaddition, we consider the lifted version of ξ ∈ L2 given byξ = (ξ0, ξ1, ξ2, . . .) = W (ξ). Due to W being isometric, wehave ‖ξ‖�2(K) = ‖ξ‖L2

, next to ‖w‖�2(K) = ‖w‖L2. Note that

we have

ξk = Mξ+k + Nwk for k ∈ N (53)

with M : Rnξ → K and N : K → K given for x ∈ Rnξ and

ω ∈ K by

(Mx)(θ) = eAθx and

(Nω)(θ) =

θ∫0

eA(θ−s)Bω(s)ds

with θ ∈ [0, h]. Note that M and N are bounded linear opera-tors. Moreover, M is invertible as a mapping fromR

nξ to im M

and its inverse is a bounded linear operator as well, since

‖Mx‖2K =

h∫0

|eAθx|2dθ

=x�h∫

0

eA�θeAθdθx ≥ ν2|x|2

where we used the fact that the Gramian∫ h0 eA

�θeAθdθ ≥ ν2I

for some ν > 0. From (53) we get for all k ∈ N that M−1(ξk −Nwk) = ξ+k and thus∣∣ξ+k ∣∣ ≤ 1

ν

[‖ξk‖K + ‖Nwk‖K

]≤ c1

(‖ξk‖K + ‖wk‖K

)for some c1 > 0, where in the latter inequality we used theboundedness of N . Using now Lemma III.5 we get for someδ > 0 that∥∥{ξ+k }k∈N∥∥�2 ≤ δmax

(‖ξ‖�2(K), ‖w‖�2(K)

)= δmax (‖ξ‖L2

, ‖w‖L2) .

Note that the constants c1 and δ do not depend on the particularξ0 and w considered. Based on internal stability of (1), theabove inequality gives∥∥{ξ+k }k∈N∥∥�2 ≤ β (max (|ξ0|, ‖w‖L2

)) (54)

for some K-function β. To transform the above bound on{ξ+k }k∈N to {ξk}k∈N, we use (16a) and the boundedness of A

and B to get for all k ∈ N that |ξk+1| ≤ c2(|ξ+k |+ ‖wk‖K) forsome c2 > 0. Again applying Lemma III.5 in combination withthe bound (54) leads to∥∥{ξk}k∈N∥∥�2 ≤ β

(max

(|ξ0|, ‖w‖�2(K)

))(55)

for some K-function β. Hence, since ξ0 and w were arbitrary,this establishes internal stability of the discrete-time system (16).

To prove the converse statement, we assume that (16) isinternally stable. Consider w ∈ L2, ξ(0) = ξ0 and τ(0) = hand the corresponding trajectory ξ of (1). Using the samenotations as above, we get from (16a) and the invertibility ofA that there exists a c3 > 0 such that∣∣ξ+k ∣∣ ≤ c3 (|ξk+1|+ ‖wk‖K) , k ∈ N. (56)

Using Lemma III.5 and internal stability of (16) guarantees theexistence of a K-function β with∥∥{ξ+k }k∈N∥∥�2 ≤ β (max (|ξ0|, ‖w‖L2

)) .

Finally, using (53) and one more time Lemma III.5 yields thedesired bound on ‖ξ‖L2

and thus the internal stability of (1).This proves the equivalence as stated in the theorem.

Moreover, note that if (1) is internally stable, due to the abovedevelopments we obtain the bound (54). Using now (53) andrealising that the operators M and N can also be consideredas bounded linear operators from R

nξ and K, respectively, toL∞[0, h], guarantee the existence of a c4 > 0 such that

‖ξk‖L∞≤ c4

(∣∣ξ+k ∣∣+ ‖wk‖K), k ∈ N. (57)


Since the right-hand side of the latter inequality can be upperbounded by using (54), we obtain that ‖ξ‖L∞ ≤ β′(max(|ξ0|,‖w‖L2

)) for some K-function β′. Since wk → 0 (in K-sense)and ξ+k →0 (cf. (56)) when k→∞, this gives based on (57) that‖ξk‖L∞

→ 0 when k → ∞ and thus we obtain limt→∞ ξ(t) =0, thereby completing the proof of the proposition.

Proof of Theorem IV.3: To prove Statement 1), let ξ0 andw = (w0, w1, . . .) ∈ �(K) result in a state sequence {ξk}k∈Nand output sequence z = (z0, z1, . . .) ∈ �(K) for system (16).Consider now the property, denoted by PK for K ∈ N, statingthat there exists (v0, v1, . . . , vK−1) ∈ KK such that (18) withinitial state ξ0 = ξ0 leads to the state trajectory {ξk}k=0,1,2,...,K

and output sequence (r0, r1, . . . , rK−1) ∈ KK satisfying (28)for k = 0, 1, 2 . . . ,K and (29) for k = 0, 1, . . . ,K − 1. Theproperty P0 obviously holds. Proceeding by induction, assumePK holds for K ∈ N. Consider now ξK = ξK and ξ+K =

φ(ξK) = ξ+K . Note that ξK+1 = Aξ+K + Bwk. We take now vKand rK according to (22) thereby satisfying (21) for k = K ,which based on (20) gives (29) for k = K . Additionally, rksatisfies (27). Moreover, from (22) and the expressions (26) and(25) it follows that:

ξK+1 = Aξ+K + Bwk

(25),(26)= Aξ+K + Bvk = Aξ+K + Bvk = ξK+1.

This shows that property PK+1 holds. Hence, using completeinduction, this proves Statement 1). Statement 2) can be provenin a similar fashion only using (23) instead of (22).

To prove Statement 3), we assume the internal stability andcontractivity of (18). First we prove the internal stability of (16).Therefore, let ξ0 and w ∈ �2(K) be given with correspondingsolution {ξk}k∈N to (16) with output sequence z ∈ �(K). Due toStatement 1) there is a v ∈ �(K) (specified through (22)) suchthat the solution {ξk}k∈N of (18) for initial state ξ0 is equalto {ξk}k∈N with output sequence r ∈ �(K). To show that v ∈�2(K) and to obtain a bound on ‖v‖�2(K) we use Θ−1 = Θ∗

(due to Θ being unitary) leading to(vkzk

)=Θ∗

(rkwk

)=

(−D∗ (I−D∗D)

12

(I−DD∗)12 D

)(rkwk

).

(58)

Since γ0 := ‖D∗‖K = ‖D‖K < 1 and c5 := ‖(I − D∗D)1/2

‖K < ∞, we get now that, for all k ∈ N, ‖vk‖K ≤ γ0‖rk‖K +c5‖wk‖K. Let us now consider the sequences v|K := (v0, v1,. . . , vK , 0, 0, 0, . . .) for K ∈ N. By using Lemma III.5 weobtain the existence of 0 ≤ γ0 < 1 and c6 > 0 such that‖v|K‖�2(K) ≤ γ0‖r|K‖�2(K) + c6‖w‖�2(K). Moreover, due tothe contractivity of (18) we also have the existence of aK-function β and 0≤γ1<1 such that ‖r|K‖�2(K)≤ β(|ξ0|) +γ1‖v|K‖�2(K). Combining these inequalities we get forall K ∈ N

‖v|K‖�2(K) ≤ γ0

[β (|ξ0|) + γ1‖v|K‖�2(K)

]+ c6‖w‖�2(K)

which gives

‖v|K‖�2(K) ≤1

1− γ0γ1γ0β (|ξ0|) +

c61− γ0γ1

‖w‖�2(K).

This proves that v ∈ �2(K) and

‖v‖�2(K) ≤1

1− γ0γ1γ0β (|ξ0|) +

c61− γ0γ1

‖w‖�2(K). (59)

Using now the internal stability of (18), we obtain∥∥{ξk}k∈N∥∥�2 =∥∥{ξk}k∈N∥∥�2 ≤β(max

(|ξ0|, ‖v‖�2(K)

))(60)

for a K-function β (independent of ξ0, w and v). Substituting(59) in (60) shows the internal stability of (16) as ξ0 and w ∈�2(K) were arbitrary.

To prove contractivity of (16), let again ξ0 and w ∈ �2(K)be given with corresponding solution {ξk}k∈N and output se-quence {zk}k∈N to (16). Following the internal stability proof,we take v ∈ �2(K) as in Statement 1) providing the solution{ξk}k∈N to (18) for initial state ξ0 and output sequence {rk}k∈Nsuch that (29) is satisfied for all k ∈ N. Note that {rk}k∈Nand {zk}k∈N are both in �2(K) due to contractivity of (18).First we will establish that there exist a positive constant ρ andK-function β1 (independent of ξ0, w) such that

‖v‖�2(K) ≥ ρ‖w‖�2(K) − β1 (|ξ0|) . (61)

Again using the identity (58), in particular, vk = −D∗rk +

(I − D∗D)1/2

wk, gives that

μ‖wk‖K ≤∥∥∥∥(I − D∗D)

12 wk

∥∥∥∥K

= ‖vk + D∗rk‖K ≤ ‖vk‖K + ‖rk‖K, k ∈ N.

Here we used that ‖D∗‖K = ‖D‖K < 1 and, moreover,since ‖D‖K < 1 we have the existence of a μ > 0 such

that for all wk ∈ K the inequality ‖(I − D∗D)1/2

wk‖K =√〈wk, (I − D∗D)wk〉K=

√‖wk‖2K − ‖Dwk‖

2

K ≥ μ‖wk‖K issatisfied. Lemma III.5 leads now to the existence of ρ1 >0 and ρ2 > 0 such that ‖w‖�2(K) ≤ ρ1‖v‖�2(K) + ρ2‖r‖�2(K).Combining the latter inequality with the contractivity of (18)gives (61) as desired. Note that (61) also leads to

‖v‖2�2(K) ≥ ρ‖w‖2�2(K) − β2 (|ξ0|) (62)

for some ρ > 0 and K-function β2. To complete the proof ofcontractivity of (16), we use again the contractivity of (18),which gives the existence of a K-function β and 0 ≤ γ1 < 1such that ‖r‖2�2(K) ≤ β(|ξ0|) + γ2

1‖v‖2�2(K), or, rewritten

‖r‖2�2(K) − ‖v‖2�2(K) ≤ β (|ξ0|) +(γ21 − 1

)‖v‖2�2(K).

Using now (29) gives that

‖z‖2�2(K) − ‖w‖2�2(K) = ‖r‖2�2(K) − ‖v‖2�2(K)

≤ β (|ξ0|) +(γ21 − 1

)‖v‖2�2(K).

Combining this inequality with (62) yields

‖z‖2�2(K) ≤ β (|ξ0|) + ‖w‖2�2(K) +(γ21 − 1

)‖v‖2�2(K)

≤ β (|ξ0|) +(1−

[1− γ2

1

]ρ)‖w‖2�2(K)

+(1− γ2

1

)β2 (|ξ0|)

which establishes the contractivity of (16) as(1−

[1− γ2

1

]ρ)< 1.

The converse statement follows in a similar manner.


Proof of Theorem IV.4: Proposition IV.1 and TheoremIV.3 show that this theorem is proven if we establish that inter-nal stability and contractivity of the system (18) is equivalent tothe internal stability and the contractivity of (30). To establishthe equivalence we will first prove the following claim.

Claim 1: The following statements are equivalent:

(1) The system (18) is internally stable and contractive.(2) The system (18) with input sequences restricted to v ∈

�2(imB∗) is internally stable and contractive.Obviously, (i) implies (ii). To show that (ii) implies (i)

note that ker B ⊕ im B∗ = K and im B∗ = (ker B)⊥ due toTheorem 1, page 57, and Theorem 3, page 157 in [37] usingthe closedness of im B∗, which is a consequence of im B∗

being a finite-dimensional subspace. Consider for system (18)an input sequence v and decompose v as v = v0 + v⊥ such thatv0 ∈ �2(ker B) and v⊥ ∈ �2(im B∗). Since v0k ∈ ker B, k ∈ N,it is obvious that for a given ξ0 the sequence v produces thesame state trajectory {ξk}k∈N and output sequence r as v⊥. Dueto the subspaces im B∗ and ker B being orthogonal, it holdsthat ‖v⊥‖�2(K) ≤ ‖v‖�2(K) and the reverse implication (ii) ⇒(i) follows as well.

Using Claim 1 we can restrict our attention to system (18)with inputs vk ∈ im B∗, k ∈ N. Take now vectors s1, . . . , sp ∈R

nξ with p = dim im B∗ < ∞ such that {B∗s1, . . . , B∗sp}

is a basis for im B∗ ⊂ K. Also consider the set of vectors{B�

d s1, . . . , B�d sp} ⊂ R

nv .Property 1: For α1, α2, . . . , αp ∈ R it holds that∥∥∥∥∥

p∑i=1

αiB∗si

∥∥∥∥∥K

=

∥∥∥∥∥p∑

i=1

αiB�d si

∥∥∥∥∥Rnv

.

This property follows from the manipulations:∥∥∥∥∥p∑

i=1

αiB∗si

∥∥∥∥∥2

K

=

⟨p∑

i=1

αiB∗si,

p∑j=1

αjB∗sj

⟩K

=

⟨B∗

(p∑

i=1

αisi

), B∗

⎛⎝ p∑

j=1

αjsj

⎞⎠⟩

K

=

⟨(p∑

i=1

αisi

), BB∗

⎛⎝ p∑

j=1

αjsj

⎞⎠⟩

Rnξ

=

⟨(p∑

i=1

αisi

), BdB

�d

⎛⎝ p∑

j=1

αjsj

⎞⎠⟩

Rnξ

=

⟨B�

d

(p∑

i=1

αisi

), B�

d

⎛⎝ p∑

j=1

αjsj

⎞⎠⟩

Rnv

=

∥∥∥∥∥p∑

i=1

αiB�d si

∥∥∥∥∥2

Rnv

.

From this property it follows that B�d s1, . . . , B

�d sp are

independent vectors, because B∗s1, . . . , B∗sp are. Moreover,

{B�d s1, . . . , B

�d sp} is a basis for im B�

d due to this indepen-dence and

rank B�d = rank Bd = dim im Bd

= dim im BdB�d = dim im BB∗

= dim im B = dim im B∗ = p.

Claim 2: The following statements are equivalent:

(a) System (18) with input sequences restricted to v ∈�2(im B∗) is internally stable and contractive.

(b) The system

ξk+1 = Aξ+k +Bdvk; ξ+k = φ(ξk); rk = Cξ+k (63)

with input sequences restricted to v ∈ �2(im B�d ) is

internally stable and contractive.

To prove that (b) implies (a) note that for each v ∈ �2(im B∗)we have that there is a unique sequence {αk}k∈N ∈ �2(R

p)such that vk =

∑pi=1 αk,iB

∗si, k ∈ N, as {B∗s1, . . . , B∗sp} is

a basis for im B∗. Then we have for all k ∈ N that

Bvk =

p∑i=1

αk,iBB∗si

=

p∑i=1

αk,iBdB�d si = Bd

(p∑

i=1

αk,iB�d si

)︸︷︷︸

=:vk

.

Clearly, v := {vk}k∈N ∈ �2(im B�d ). Due to statement (b)

and Property 1 (and thus ‖vk‖K = ‖vk‖Rnv and ‖v‖�2(K) =‖v‖�2(K)), we have (a) as the system (18) with input v andinitial state ξ0 and the system (63) with input v (with the samenorm as v) and the same initial state produce the same statetrajectory {ξk}k∈N and output response {rk}k∈N. The reverseimplication (a) ⇒ (b) follows in a similar manner using that{B�

d s1, . . . , B�d sp} is a basis for im B�

d .Claim 3: The following statements are equivalent:

(A) The system (63) with input sequences restricted to v ∈�2(im B�

d ) is internally stable and contractive.(B) The system (63) is internally stable and contractive.

Claim 3 can be proven analogously to Claim 1 usingkerBd ⊕ im B�

d = Rnv .

Combining now Claims 1, 2, and 3 yields that the followingstatements are equivalent:

(1) The system (18) is internally stable and contractive.(2) The system (63) is internally stable and contractive.

Considering now that an output sequence r of system (63)(for some ξ0 and v ∈ �2) satisfies

‖r‖2�2(K)=

∞∑k=0

‖rk‖2K =

∞∑k=0

⟨Cξ+k , C ξ+k

⟩K

=

∞∑k=0

⟨ξ+k , C

∗Cξ+k⟩R

nξ =

∞∑k=0

⟨ξ+k , C�

d Cdξ+k

⟩R

nξ

=

∞∑k=0

⟨Cdξ

+k ,Cdξ

+k

⟩Rnr

=

∞∑k=0

∥∥Cdξ+k

∥∥2Rnr

=‖r‖2�2(Rnr )

where r is the output sequence of (30) (for the same ξ0 andv ∈ �2), it follows that (18) is internally stable and contractiveif and only if (30) is.

Proof of Theorem VI.3: In the proof of Statement 1)we will use for τ ∈ R≥0 the operator DA,B,C,D

τ :L2[0, τ ]→L2[0, τ ] defined through (DA,B,C,D

τ w)(t)=∫ t0CeA(t−η)Bw(η)dη+Dw(t) with t ∈ [0, τ ] for w ∈ L2[0, τ ].

We prove now first Statement 1) for the case D = 0 byconsidering Dτ := DA,B,C,0

τ . Note that Dh = D. In [38] it is


proven that I − D∗τ Dτ is invertible if and only if F11(τ) is

invertible, see also the discussion on page 432 of [17]. “⇒”Since ‖D‖K < 1 implies ‖Dτ‖L2[0,τ ]

< 1 for all τ ∈ [0, h] and

thus the invertibility of I − D∗τ Dτ for all τ ∈ [0, h], we can

use the above mentioned result in [38] to get for all τ ∈ [0, h]the invertibility of F11(τ). “⇐” We show the conversestatement by assuming ‖D‖K ≥ 1. Since τ �→ ‖Dτ‖L2[0,τ ] is

a continuous function and limτ↓0 ‖Dτ‖L2[0,τ ]= 0, ‖D‖K ≥ 1

implies the existence of a τ ∈ [0, h] such that ‖Dτ‖L2[0,τ ] = 1.

The latter condition results in I − D∗τ Dτ not being invertible

and thus that F11(τ) is not invertible. As a consequence,Assumption VI.1 is not true, thereby completing the proof forthe case D = 0.

The case D �= 0 follows from the case D = 0 by using astandard (pointwise) loop-shifting argument in a similar way asdone in Section IV-B leading to the equivalence of the followingstatements:

• ‖DA,B,C,Dh ‖K < 1.

• ‖D‖=√λmax(D�D)<1 and ‖DAl,Bl,Cl,0

h ‖K<1 withAl=A+BMD�C, Bl = BM1/2 and Cl = L1/2C.

By applying now Statement 1) for D = 0 to the latter estab-lishes Statement 1) also for D �= 0.

Statement 2) follows by proving the identities (see (33), (34),and (49)):

F−111 =Q11(h)

−� (64a)

−F−111 F12 =Q21(h)Q11(h)

−1 (64b)

F21F−111 = −Q11(h)

−1Q12(h). (64c)

To prove these identities, first we define E := E1 and observe

that for T =

(0 II 0

)we get E = THT−1 and T = T−1

such that eEt = TeHtT−1. DefiningG(t) := eHt and partition-ing this similarly as E we obtain from eEt = TeHtT that

Q11(t) =G22(t); Q12(t) = G21(t) (65a)Q21(t) =G12(t); Q22(t) = G11(t). (65b)

Due to F (t) satisfying F�(t)ΩF (t) = Ω for all t ∈ R with

Ω =

(0 I−I 0

)(see proof Theorem III.2 in [2]), we obtain

ΩF (t) = [F�(t)]−1Ω. Combining the latter identity with

[F�(t)

]−1= F�(−t) = [eHt]

�=

(G�

11(t) G�21(t)

G�12(t) G�

22(t)

)

leads to ΩF (t) = G�Ω. This gives

F21 = −G21(h)�; F22 = G11(h)

� (66a)

F11 =G22(h)�; F12 = −G12(h)

�. (66b)

Combining (65) and (66) gives F−111 = Q11(h)

−�

−F−111 F12 =Q11(h)

−�Q21(h)�

=(Q21(h)Q11(h)

−1)�

= Q21(h)Q11(h)−1

F21F−111 = −Q12(h)

�Q11(h)−�

= −[Q11(h)

−1Q12(h)]�

= −Q11(h)−1Q12(h).

These are the desired inequalities in (49).To prove Statement 3) we assume that the hypotheses of

Proposition VI.2 hold, i.e., Assumption VI.1 holds and there area matrix Ph � 0 and scalars μi ≥ 0, i ∈ {1, 2}, satisfying theLMIs (47). Due to Statement 1) this guarantees that ‖D‖K < 1.

To link the feasibility of the LMIs in Proposition VI.2 tothe feasibility of the LMIs in Theorem V.1 it is important tonote that according to [39, Sec. 2.6.3] the implication ξ�Xiξ ≥0 ⇒ ξ�Qξ ≤ 0 for i = 1, 2, . . . , N1 yields the existence ofζi, i = 1, 2, . . . , N1, such that −Q � ζiXi, i = 1, 2, . . . , N1.Similarly, since ξ�Xiξ ≥ 0 ⇒ ξ�Qξ ≥ 0 for i=N1+1,. . . , N there exist ζi, i = N1 + 1, . . . , N , such that Q � ζiXi,i = N1 + 1, . . . , N . Also note that due to Statement 2) thePWL systems used in Proposition VI.2 and Theorem V.1 areessentially the same. As such, if Ph � 0, and μi ≥ 0, i ∈{1, 2} satisfy (47), it follows that Pi = Ph and κi = 0, i =1, 2, . . . , N , βi,j = 0, i, j = 1, 2, . . . , N , and μi,j = ζiμ1 fori = N1 + 1, . . . , N , j = 1, 2, . . . , N , and μi,j = ζiμ2 for i =1, . . . , N1, j = 1, 2, . . . , N , form a solution to the LMIs (39) .This completes the proof.

ACKNOWLEDGMENT

The authors wish to thank B. van Loon for his help inpreparing the numerical examples.

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W. P. M. H. Heemels (F’16) received the M.Sc.degree (with highest honors) in mathematics andthe Ph.D. degree (with highest honors) in controltheory from the Eindhoven University of Technology(TU/e), Eindhoven, the Netherlands, in 1995 and1999, respectively.

From 2000 to 2004, he was with the ElectricalEngineering Department, TU/e, as an Assistant Pro-fessor and from 2004 to 2006 with the EmbeddedSystems Institute (ESI) as a Research Fellow. Since2006, he has been with the Department of Mechan-

ical Engineering, TU/e, where he is currently a Full Professor in the ControlSystems Technology Group. He held visiting research positions at the SwissFederal Institute of Technology (ETH), Zurich, Switzerland (2001) and at theUniversity of California at Santa Barbara (2008). In 2004, he was also atthe Research and Development Laboratory, Océ, Venlo, the Netherlands. Hiscurrent research interests include hybrid and cyber-physical systems, networkedand event-triggered control systems and constrained systems including modelpredictive control.

Dr. Heemels received the VICI Grant from NWO (The Netherlands Organ-isation for Scientific Research) and STW (Dutch Technology Foundation). Heserved/s on the editorial boards of Automatica, Nonlinear Analysis: HybridSystems, Annual Reviews in Control, and the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL.

G. E. Dullerud (F’08) was born in Oslo, Norway,in 1966. He received the B.A.Sc. degree in engi-neering science and the M.A.Sc. degree in electricalengineering from the University of Toronto, Toronto,ON, Canada, in 1988 and 1990, respectively, and thePh.D. degree in engineering from the University ofCambridge, Cambridge, U.K., in 1994.

Since 1998, he has been a faculty member in Me-chanical Science and Engineering at the Universityof Illinois, Urbana-Champaign, where he is currentlya Professor. He is the Director of the Decision and

Control Laboratory of the Coordinated Science Laboratory. He has held visitingpositions in electrical engineering at KTH, Stockholm, Sweden, in 2013, and inAeronautics and Astronautics at Stanford University during 2005–2006. From1996 to 1998, he was an Assistant Professor in Applied Mathematics at theUniversity of Waterloo, Waterloo, ON, Canada. He was a Research Fellowand Lecturer in the Control and Dynamical Systems Department, CaliforniaInstitute of Technology, in 1994 and 1995. He has published two books:A Course in Robust Control Theory (New York: Springer, 2000) and Controlof Uncertain Sampled-data Systems (Boston, MA: Birkhauser 1996). He iscurrently Associate Editor of the SIAM Journal on Control and Optimization,and served in a similar role for Automatica. His areas of current research inter-ests include games and networked control, robotic vehicles, hybrid dynamicalsystems, and cyber-physical systems security.

Dr. Dullerud received the National Science Foundation CAREER Award in1999, and the Xerox Faculty Research Award at UIUC in 2005. He became anASME Fellow in 2011. He was an Associate Editor of the IEEE TRANSAC-TIONS ON AUTOMATIC CONTROL.

A. R. Teel (F’02) received the A.B. degree in engi-neering sciences from Dartmouth College, Hanover,NH, in 1987, and the M.S. and Ph.D. degrees in elec-trical engineering from the University of California,Berkeley, in 1989 and 1992, respectively.

He was a Postdoctoral Fellow at the Ecole desMines de Paris, Fontainebleau, France. In 1992he joined the faculty of the Electrical EngineeringDepartment, University of Minnesota, Minneapolis,where he was an Assistant Professor until 1997. Sub-sequently, he joined the faculty of the Electrical and

Computer Engineering Department, University of California, Santa Barbara,where he is currently a Professor. He is an Area Editor for Automatica. Hisresearch interests are in nonlinear and hybrid dynamical systems, with a focuson stability analysis and control design.

Dr. Teel received the NSF Research Initiation and CAREER Awards, the1998 IEEE Leon K. Kirchmayer Prize Paper Award, the 1998 George S. AxelbyOutstanding Paper Award, the first SIAM Control and Systems Theory Prize in1998, the 1999 Donald P. Eckman Award and the 2001 O. Hugo Schuck BestPaper Award, and the 2010 IEEE Control Systems Magazine Outstanding PaperAward. He is a Fellow of IFAC.

l -gain analysis for a class of hybrid systems with

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